Finite Math Examples

$3 x + 2 y = 5$ , $6 x - 2 y = 5$

Step 1

Solve for $x$ in $3 x + 2 y = 5$ .

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Step 1.1

Subtract $2 y$ from both sides of the equation.

$3 x = 5 - 2 y$

$6 x - 2 y = 5$

Step 1.2

Divide each term in $3 x = 5 - 2 y$ by $3$ and simplify.

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Step 1.2.1

Divide each term in $3 x = 5 - 2 y$ by $3$ .

$\frac{3 x}{3} = \frac{5}{3} + \frac{- 2 y}{3}$

$6 x - 2 y = 5$

Step 1.2.2

Simplify the left side.

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Step 1.2.2.1

Cancel the common factor of $3$ .

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Step 1.2.2.1.1

Cancel the common factor.

$\frac{3 x}{3} = \frac{5}{3} + \frac{- 2 y}{3}$

$6 x - 2 y = 5$

Step 1.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{5}{3} + \frac{- 2 y}{3}$

$6 x - 2 y = 5$

$x = \frac{5}{3} + \frac{- 2 y}{3}$

$6 x - 2 y = 5$

$x = \frac{5}{3} + \frac{- 2 y}{3}$

$6 x - 2 y = 5$

Step 1.2.3

Simplify the right side.

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Step 1.2.3.1

Move the negative in front of the fraction.

$x = \frac{5}{3} - \frac{2 y}{3}$

$6 x - 2 y = 5$

$x = \frac{5}{3} - \frac{2 y}{3}$

$6 x - 2 y = 5$

$x = \frac{5}{3} - \frac{2 y}{3}$

$6 x - 2 y = 5$

$x = \frac{5}{3} - \frac{2 y}{3}$

$6 x - 2 y = 5$

Step 2

Replace all occurrences of $x$ with $\frac{5}{3} - \frac{2 y}{3}$ in each equation.

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Step 2.1

Replace all occurrences of $x$ in $6 x - 2 y = 5$ with $\frac{5}{3} - \frac{2 y}{3}$ .

$6 (\frac{5}{3} - \frac{2 y}{3}) - 2 y = 5$

$x = \frac{5}{3} - \frac{2 y}{3}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify $6 (\frac{5}{3} - \frac{2 y}{3}) - 2 y$ .

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Step 2.2.1.1

Simplify each term.

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Step 2.2.1.1.1

Apply the distributive property.

$6 (\frac{5}{3}) + 6 (- \frac{2 y}{3}) - 2 y = 5$

$x = \frac{5}{3} - \frac{2 y}{3}$

Step 2.2.1.1.2

Cancel the common factor of $3$ .

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Step 2.2.1.1.2.1

Factor $3$ out of $6$ .

$3 (2) (\frac{5}{3}) + 6 (- \frac{2 y}{3}) - 2 y = 5$

$x = \frac{5}{3} - \frac{2 y}{3}$

Step 2.2.1.1.2.2

Cancel the common factor.

$3 \cdot (2 (\frac{5}{3})) + 6 (- \frac{2 y}{3}) - 2 y = 5$

$x = \frac{5}{3} - \frac{2 y}{3}$

Step 2.2.1.1.2.3

Rewrite the expression.

$2 \cdot 5 + 6 (- \frac{2 y}{3}) - 2 y = 5$

$x = \frac{5}{3} - \frac{2 y}{3}$

$2 \cdot 5 + 6 (- \frac{2 y}{3}) - 2 y = 5$

$x = \frac{5}{3} - \frac{2 y}{3}$

Step 2.2.1.1.3

Multiply $2$ by $5$ .

$10 + 6 (- \frac{2 y}{3}) - 2 y = 5$

$x = \frac{5}{3} - \frac{2 y}{3}$

Step 2.2.1.1.4

Cancel the common factor of $3$ .

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Step 2.2.1.1.4.1

Move the leading negative in $- \frac{2 y}{3}$ into the numerator.

$10 + 6 (\frac{- 2 y}{3}) - 2 y = 5$

$x = \frac{5}{3} - \frac{2 y}{3}$

Step 2.2.1.1.4.2

Factor $3$ out of $6$ .

$10 + 3 (2) (\frac{- 2 y}{3}) - 2 y = 5$

$x = \frac{5}{3} - \frac{2 y}{3}$

Step 2.2.1.1.4.3

Cancel the common factor.

$10 + 3 \cdot (2 (\frac{- 2 y}{3})) - 2 y = 5$

$x = \frac{5}{3} - \frac{2 y}{3}$

Step 2.2.1.1.4.4

Rewrite the expression.

$10 + 2 (- 2 y) - 2 y = 5$

$x = \frac{5}{3} - \frac{2 y}{3}$

$10 + 2 (- 2 y) - 2 y = 5$

$x = \frac{5}{3} - \frac{2 y}{3}$

Step 2.2.1.1.5

Multiply $- 2$ by $2$ .

$10 - 4 y - 2 y = 5$

$x = \frac{5}{3} - \frac{2 y}{3}$

$10 - 4 y - 2 y = 5$

$x = \frac{5}{3} - \frac{2 y}{3}$

Step 2.2.1.2

Subtract $2 y$ from $- 4 y$ .

$10 - 6 y = 5$

$x = \frac{5}{3} - \frac{2 y}{3}$

$10 - 6 y = 5$

$x = \frac{5}{3} - \frac{2 y}{3}$

$10 - 6 y = 5$

$x = \frac{5}{3} - \frac{2 y}{3}$

$10 - 6 y = 5$

$x = \frac{5}{3} - \frac{2 y}{3}$

Step 3

Solve for $y$ in $10 - 6 y = 5$ .

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Step 3.1

Move all terms not containing $y$ to the right side of the equation.

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Step 3.1.1

Subtract $10$ from both sides of the equation.

$- 6 y = 5 - 10$

$x = \frac{5}{3} - \frac{2 y}{3}$

Step 3.1.2

Subtract $10$ from $5$ .

$- 6 y = - 5$

$x = \frac{5}{3} - \frac{2 y}{3}$

$- 6 y = - 5$

$x = \frac{5}{3} - \frac{2 y}{3}$

Step 3.2

Divide each term in $- 6 y = - 5$ by $- 6$ and simplify.

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Step 3.2.1

Divide each term in $- 6 y = - 5$ by $- 6$ .

$\frac{- 6 y}{- 6} = \frac{- 5}{- 6}$

$x = \frac{5}{3} - \frac{2 y}{3}$

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Cancel the common factor of $- 6$ .

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Step 3.2.2.1.1

Cancel the common factor.

$\frac{- 6 y}{- 6} = \frac{- 5}{- 6}$

$x = \frac{5}{3} - \frac{2 y}{3}$

Step 3.2.2.1.2

Divide $y$ by $1$ .

$y = \frac{- 5}{- 6}$

$x = \frac{5}{3} - \frac{2 y}{3}$

$y = \frac{- 5}{- 6}$

$x = \frac{5}{3} - \frac{2 y}{3}$

$y = \frac{- 5}{- 6}$

$x = \frac{5}{3} - \frac{2 y}{3}$

Step 3.2.3

Simplify the right side.

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Step 3.2.3.1

Dividing two negative values results in a positive value.

$y = \frac{5}{6}$

$x = \frac{5}{3} - \frac{2 y}{3}$

$y = \frac{5}{6}$

$x = \frac{5}{3} - \frac{2 y}{3}$

$y = \frac{5}{6}$

$x = \frac{5}{3} - \frac{2 y}{3}$

$y = \frac{5}{6}$

$x = \frac{5}{3} - \frac{2 y}{3}$

Step 4

Replace all occurrences of $y$ with $\frac{5}{6}$ in each equation.

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Step 4.1

Replace all occurrences of $y$ in $x = \frac{5}{3} - \frac{2 y}{3}$ with $\frac{5}{6}$ .

$x = \frac{5}{3} - \frac{2 (\frac{5}{6})}{3}$

$y = \frac{5}{6}$

Step 4.2

Simplify the right side.

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Step 4.2.1

Simplify $\frac{5}{3} - \frac{2 (\frac{5}{6})}{3}$ .

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Step 4.2.1.1

Combine the numerators over the common denominator.

$x = \frac{5 - 2 (\frac{5}{6})}{3}$

$y = \frac{5}{6}$

Step 4.2.1.2

Simplify each term.

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Step 4.2.1.2.1

Cancel the common factor of $2$ .

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Step 4.2.1.2.1.1

Factor $2$ out of $- 2$ .

$x = \frac{5 + 2 (- 1) (\frac{5}{6})}{3}$

$y = \frac{5}{6}$

Step 4.2.1.2.1.2

Factor $2$ out of $6$ .

$x = \frac{5 + 2 \cdot (- 1 \frac{5}{2 \cdot 3})}{3}$

$y = \frac{5}{6}$

Step 4.2.1.2.1.3

Cancel the common factor.

$x = \frac{5 + 2 \cdot (- 1 \frac{5}{2 \cdot 3})}{3}$

$y = \frac{5}{6}$

Step 4.2.1.2.1.4

Rewrite the expression.

$x = \frac{5 - 1 (\frac{5}{3})}{3}$

$y = \frac{5}{6}$

$x = \frac{5 - 1 (\frac{5}{3})}{3}$

$y = \frac{5}{6}$

Step 4.2.1.2.2

Rewrite $- 1 (\frac{5}{3})$ as $- (\frac{5}{3})$ .

$x = \frac{5 - \frac{5}{3}}{3}$

$y = \frac{5}{6}$

$x = \frac{5 - \frac{5}{3}}{3}$

$y = \frac{5}{6}$

Step 4.2.1.3

To write $5$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$x = \frac{5 \cdot \frac{3}{3} - \frac{5}{3}}{3}$

$y = \frac{5}{6}$

Step 4.2.1.4

Combine $5$ and $\frac{3}{3}$ .

$x = \frac{\frac{5 \cdot 3}{3} - \frac{5}{3}}{3}$

$y = \frac{5}{6}$

Step 4.2.1.5

Combine the numerators over the common denominator.

$x = \frac{\frac{5 \cdot 3 - 5}{3}}{3}$

$y = \frac{5}{6}$

Step 4.2.1.6

Simplify the numerator.

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Step 4.2.1.6.1

Multiply $5$ by $3$ .

$x = \frac{\frac{15 - 5}{3}}{3}$

$y = \frac{5}{6}$

Step 4.2.1.6.2

Subtract $5$ from $15$ .

$x = \frac{\frac{10}{3}}{3}$

$y = \frac{5}{6}$

$x = \frac{\frac{10}{3}}{3}$

$y = \frac{5}{6}$

Step 4.2.1.7

Multiply the numerator by the reciprocal of the denominator.

$x = \frac{10}{3} \cdot \frac{1}{3}$

$y = \frac{5}{6}$

Step 4.2.1.8

Multiply $\frac{10}{3} \cdot \frac{1}{3}$ .

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Step 4.2.1.8.1

Multiply $\frac{10}{3}$ by $\frac{1}{3}$ .

$x = \frac{10}{3 \cdot 3}$

$y = \frac{5}{6}$

Step 4.2.1.8.2

Multiply $3$ by $3$ .

$x = \frac{10}{9}$

$y = \frac{5}{6}$

$x = \frac{10}{9}$

$y = \frac{5}{6}$

$x = \frac{10}{9}$

$y = \frac{5}{6}$

$x = \frac{10}{9}$

$y = \frac{5}{6}$

$x = \frac{10}{9}$

$y = \frac{5}{6}$

Step 5

The solution to the system is the complete set of ordered pairs that are valid solutions.

$(\frac{10}{9}, \frac{5}{6})$

Step 6

The result can be shown in multiple forms.

Point Form:

$(\frac{10}{9}, \frac{5}{6})$

Equation Form:

$x = \frac{10}{9}, y = \frac{5}{6}$

Step 7